Leftist tree/ leftist heap

Assoume you give away points to the 100 best Students.

Assume you have n > 100 students and all studnets have a diffrent amount of points. Let 0 be the middle from the scale. If the point are lower, the Student is a bad one, if its a higher amount of points the student is a good one. Also assume the Datatype where the students are saved is a leftist tree/heap but only with the Operations:

init(k) -> O(1) makes a tree with with the key "k"

insert(D,k) -> O(log n) inserts a key "k" into the tree D

delete-min(D) -> O(log n) deletes the smalles element from tree D

relocate(D,k,y) -> O(n) first delete(D,k) than changes k to y and insert(D,k) after that you replaced the key "k" by "y"

merge(D1,D2) -> O(log n) merges two trees

O(x) -> Runtime x // Big O

The key x of a student k is the value from the points. If the student is lower in the tree from k in D, so k gets more points, that means the student at the top root gets the lowest points.

i) Assume Studnet k is a better student than k'. Does k 100% gets more points than k'? explain why/why not?

ii) Assume a imp offers you a deal.He offers you to lower your Distancevalue by 1 if its >1. Wich operations do he have to use and why you should take this offer/why not?

iii)Assume that you want to get the 100 baddest students. because you want give them any points. Describe a algorithm by words or pseudocode how you can get them in a runtime of O(log n) explain why the runtime is respected.

iv) now you want to get the best students in a runtime of O(n log n). again describe the algorithm by words and why the runtime is respected.